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Q13:
Find the direction cosines of the vector joining the points A (1, 2, β3) and B (β1, β2, 1), directed from A to B.
Solution :
Step 1: Determining the Vector Component Form
To find the vector $\vec{AB}$ directed from point $A(x_1, y_1, z_1)$ to $B(x_2, y_2, z_2)$, we subtract the coordinates of $A$ from $B$: $\vec{AB} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$ [By the definition of a position vector between two points].
Given $A(1, 2, -3)$ and $B(-1, -2, 1)$:
- $\Delta x = -1 - 1 = -2$
- $\Delta y = -2 - 2 = -4$
- $\Delta z = 1 - (-3) = 4$
Thus, $\vec{AB} = -2\hat{i} - 4\hat{j} + 4\hat{k}$.
Step 2: Calculating the Magnitude of Vector $\vec{AB}$
The magnitude $|\vec{AB}|$ is given by the square root of the sum of the squares of the components: $|\vec{AB}| = \sqrt{a^2 + b^2 + c^2}$ [Per the magnitude formula for a 3D vector].
$|\vec{AB}| = \sqrt{(-2)^2 + (-4)^2 + 4^2}$
$|\vec{AB}| = \sqrt{4 + 16 + 16} = \sqrt{36} = 6$ units.
Step 3: Calculating Direction Cosines
The direction cosines $(l, m, n)$ are defined as the ratio of each component to the magnitude of the vector: $l = \frac{a}{|\vec{r}|}, m = \frac{b}{|\vec{r}|}, n = \frac{c}{|\vec{r}|}$
- $l = \frac{-2}{6} = -\frac{1}{3}$
- $m = \frac{-4}{6} = -\frac{2}{3}$
- $n = \frac{4}{6} = \frac{2}{3}$
Geometric Representation
Step 4: Verification of Consistency
Check the identity $l^2 + m^2 + n^2 = 1$ [Fundamental property of direction cosines]:
$(-\frac{1}{3})^2 + (-\frac{2}{3})^2 + (\frac{2}{3})^2 = \frac{1}{9} + \frac{4}{9} + \frac{4}{9} = \frac{9}{9} = 1$.
The calculation is verified as accurate.
Final Solution: The direction cosines of vector $\vec{AB}$ are $\left( -\frac{1}{3}, -\frac{2}{3}, \frac{2}{3} \right)$.
More Questions from Class 12 Mathematics Vector Algebra EXERCISE 10.2
- Q1(i): Compute the magnitude of the following vectors: $\vec{a} = \hat{i} + \hat{j} + \hat{k}$
- Q1(ii): Compute the magnitude of the following vectors: $\vec{b} = 2\hat{i} - 7\hat{j} - 3\hat{k}$
- Q1(iii): Compute the magnitude of the following vectors: $\vec{c} = \frac{1}{3}\hat{i} + \frac{1}{3}\hat{j} - \frac{1}{3}\hat{k}$
- Q2: Write two different vectors having same magnitude.
- Q3: Write two different vectors having same direction.
- Q4: Find the values of $x$ and $y$ so that the vectors $2\hat{i} + 3\hat{j}$ and $x\hat{i} + y\hat{j}$ are equal.
- Q5: Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (β 5, 7).
- Q6: Find the sum of the vectors $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$, $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$ and $\vec{c} = \hat{i} - 6\hat{j} - 7\hat{k}$.
- Q7: Find the unit vector in the direction of the vector $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$.
- Q8: Find the unit vector in the direction of vector $\vec{PQ}$, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.
- Q9: For given vectors, $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{b} = -\hat{i} + \hat{j} - \hat{k}$, find the unit vector in the direction of the vector $\vec{a} + \vec{b}$.
- Q10: Find a vector in the direction of vector $5\hat{i} - \hat{j} + 2\hat{k}$ which has magnitude 8 units.
- Q11: Show that the vectors $2\hat{i} - 3\hat{j} + 4\hat{k}$ and $-4\hat{i} + 6\hat{j} - 8\hat{k}$ are collinear.
- Q12: Find the direction cosines of the vector $\hat{i} + 2\hat{j} + 3\hat{k}$.
- Q14: Show that the vector $\hat{i} + \hat{j} + \hat{k}$ is equally inclined to the axes OX, OY and OZ.
- Q15(i): Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$ respectively, in the ratio 2 : 1 (i) internally
- Q15(ii): Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$ respectively, in the ratio 2 : 1 (ii) externally
- Q16: Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, β2).
- Q17: Show that the points A, B and C with position vectors, $\vec{a} = 3\hat{i} - 4\hat{j} - 4\hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - 3\hat{j} - 5\hat{k}$, respectively form the vertices of a right angled triangle.
- Q18: In triangle ABC (Fig 10.18), which of the following is not true:
- Q19: If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then which of the following are incorrect:
CBSE Solutions for Class 12 Mathematics Vector Algebra
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